严格来讲，这两句子并不相等：

Try as he might,  he couldn't open the door.

However hard he tried, he couldn't open the door.

1. 第一句意思：虽然可能尝试过，但他没有打开门。没有 hard 的try，仅仅是“尝试”。第二句意思：无论多么努力过，他也没有打开门。try hard 才是“努力”。

2. 第一句用 might，仅表示“可能”，但不一定实施。第二句则是一般过去时 tried hard，说明已经努力过。

3. 这两句虽然都是让步，但程度大不同。

第一句的结构：动词原形（若有宾语或状语，则一并提前）+ as + 主语 + 助动词（do, did）/情态动词。更多例句：

Work hard as he will, he never seems able to do the work satisfactorily. 虽然他工作很努力，但总不能让人满意。

Lose money as I did, we got a lot of experience. 钱虽然丢了，我们却得到了许多经验。

Fail in the election as he did, he became famous for his fiery speech against slavery. 尽管落选了，但他却以其反对奴隶制的激烈演说而出了名。

Dislike him as we may, we must acknowledge his greatness. 尽管我们不喜欢他，但必须承认他的伟大。